Kristen invests $ 5745 in a bank. The bank 6.5% interest compounded monthly. How long must she leave the money in the bank for it to double? Round to the nearest tenth of a year. Show your work.
How long will it take to triple? Round to nearest tenth of a year. Show your work.
Kristen has a choice to invest her money at 6.5% interest compounded monthly for 5 years or invest her money compounding quarterly at a rate of 6.75% for 5 years. What option would be best for Kristen? Explain and show your work.

Respuesta :

frika

Answer:

1. 10.7 years

2. 17.0 years

3. 2nd option

Step-by-step explanation:

Use formula for compounded interest

[tex]A=P\cdot \left(1+\dfrac{r}{n}\right)^{nt},[/tex]

where

A is final value, P is initial value, r is interest rate (as decimal), n is number of periods and t is number of years.

In your case,

1. P=$5745, A=2P=$11490, n=12 (compounded monthly), r=0.065 (6.5%) and t is unknown. Then

[tex]11490=5745\cdot \left(1+\dfrac{0.065}{12}\right)^{12t},\\ \\2=(1.0054)^{12t},\\ \\12t=\log_{1.0054}2,\\ \\t=\dfrac{1}{12}\log_ {1.0054}2\approx 10.7\ years.[/tex]

2. P=$5745, A=3P=$17235, n=12 (compounded monthly), r=0.065 (6.5%) and t is unknown. Then


[tex]17235=5745\cdot \left(1+\dfrac{0.065}{12}\right)^{12t},[/tex]

[tex]3=(1.0054)^{12t},[/tex]

[tex]12t=\log_{1.0054}3,[/tex]

[tex]t=\dfrac{1}{12}\log_{1.0054}3\approx 17.0\ years.[/tex]

3. 1 choice: P=$5745, n=12 (compounded monthly), r=0.065 (6.5%), t=5 years and A is unknown. Then

[tex]A=5745\cdot \left(1+\dfrac{0.065}{12}\right)^{12\cdot 5},\\ \\A=5745\cdot (1.0054)^{60}\approx \$7936.39.[/tex]

2 choice: P=$5745, n=4 (compounded monthly), r=0.0675 (6.75%), t=5 years and A is unknown. Then

[tex]A=5745\cdot \left(1+\dfrac{0.0675}{4}\right)^{4\cdot 5},\\ \\A=5745\cdot (1.0169)^{20}\approx \$8032.58.[/tex]

The best will be 2nd option, because $8032.58>$7936.39